A mathematical proof is an inferential argument for a mathematical statement, showing that the stated assumptions logically guarantee the conclusion. The argument may use other previously established statements, such as theorems; but every proof can, in principle, be constructed using only certain basic or original assumptions known as axioms, along with the accepted rules of inference. Proofs are examples of exhaustive deductive reasoning which establish logical certainty, to be distinguished from empirical arguments or non-exhaustive inductive reasoning which establish "reasonable expectation". Presenting many cases in which the statement holds is not enough for a proof, which must demonstrate that the statement is true in all possible cases. An unproven proposition that is believed to be true is known as a conjecture, or a hypothesis if frequently used as an assumption for further mathematical work.Proofs employ logic expressed in mathematical symbols, along with natural language which usually admits some ambiguity. In most mathematical literature, proofs are written in terms of rigorous informal logic. Purely formal proofs, written fully in symbolic language without the involvement of natural language, are considered in proof theory. The distinction between formal and informal proofs has led to much examination of current and historical mathematical practice, quasi-empiricism in mathematics, and so-called folk mathematics, oral traditions in the mainstream mathematical community or in other cultures. The philosophy of mathematics is concerned with the role of language and logic in proofs, and mathematics as a language.
Homework Statement
Take the expression 21.11 (pictured below, specifically the bottom one) for the electric field above the center of a uniformly charged disk with radius R and surface charge density σ, and show that when one is very far from the disk, the field decreases with the same square...
1. The question is. Show that if |nx| <1, the following is exact up to (and including) the x^2 order. The hint giving says to use the Taylor Expansion for both sides of the equation2. (1+x)^n = e^n(x-(1/2)x^2) ; the n(x-(1/2)x^2) is all an exponent3. My first attempt was to take the taylor...
Hi. In the attached proof for Lorentz transformation for momentum http://www.colorado.edu/physics/phys2170/phys2170_sp07/downloads/lorentz_transformation_E_p.pdf, there is this step that I don't understand:
1/√1-u'2/c2 = γ(1-vux/c2)/√1-u2/c2
Can someone explain how they derived this? Thanks! :)
Homework Statement
If gcd(f(x),g(x)) = 1 and m,n ∈ ℕ, show that gcd(f(x)^m, g(x)^n) = 1.
Homework EquationsThe Attempt at a Solution
So I had previously proved this for non-polynomials:
gcd(a,b)=1
then gcd(a^n,b^n)=1
Proof: a = p1*p2*...*pn
b = p1*p2*...*pm
then
a^n = p1^n*p2^n*...*pn^n...
I'm going through Bishop and Goldberg's "Tensor Analysis on Manifolds" right now and I'm stuck in Chapter 0. :H They give a proof of the statement "A compact subset of a Hausdorff space is closed" that I can't seem to wrap my head around. I'm reprinting the proof here:
"Suppose that A is a...
Homework Statement
Attached is the problem.
I didn't find anyway to apply the hamming distance to this problem, but I hope that at least this is close to show something.
Homework EquationsThe Attempt at a Solution
Lets consider Rn over Z 2 n, so the basis of R n under Z 2 is
(0,0,………0 )all the...
Homework Statement
Show that all square matrix (A whatever) can be written as the sum of a symmetric matrix and a anti symmetric matrix.
Homework Equations
I think this relation might be relevant : $$
A=\frac{1}{2}*(A+A^{T})+\frac{1}{2}*(A-A^{T})
$$
The Attempt at a Solution
I know that we...
Using these equations I am about to prove that photons have a rest mass of zero (mathematically)
________________________________________________________________________________________
E=hc/λ Photon Energy Equation
E2=(pc+mc2)2 Mass-Energy Equivalence with Momentum Equation
p=h/λ Momentum...
I'd like to discuss the question in the title, following up on my remark quoted below.
Note that I don't want to repeat the discussion in
https://www.physicsforums.com/threads/tracks-in-particle-detectors-and-quantum-paths.758778
so maybe reread that one first!
The traditional analysis is...
Homework Statement
Let r be an element of an integral domain R such that r^2 = r. Show that either r = 0_R or 1_R
Homework Equations
integral domain means no zero divisors.
The Attempt at a Solution
This is fundamental as 0 and 1 solve r^2 = r and are the only solutions.
However, I'm not...
Homework Statement
Let T:R-> S be a homomorphism of rings. Show that T(0_r) = 0_s.
Homework EquationsThe Attempt at a Solution
First off, the terminology used is kinda confusing. I take 0_r to be the zero in R. Is this correct? For some reason I recall my teacher quickly saying that it was...
From Baby Rudin
"Thm: Let P be a non-empty, perfect subset of R^k. Then P is uncountable.
Pf: Since P has limit points, P must be infinite. Suppose P is countable, list the point of P {x1 ...xn }. Construct a sequence of nbhds. as follows. Let V1 be any nbhd of x1 . Suppose Vn has been...
Homework Statement
Let ##\sigma_4## denote the group of permutations of ##\{1,2,3,4\}## and consider the following elements in ##\sigma_4##:
##x=\bigg(\begin{matrix}1&&2&&3&&4\\2&&1&&4&&3\end{matrix}\bigg);~~~~~~~~~y=\bigg(\begin{matrix}1&&2&&3&&4\\3&&4&&1&&2\end{matrix}\bigg)##...
I've just encountered this somewhere and I need some sort of formal proof for why a continuous function ##f(x)## can equal zero because its integral is zero. Are there any out there? I've seen similar forum posts on places like Stack Exchange and one here, but I can't exactly follow the logic...
In Andrew McInerney's book: First Steps in Differential Geometry, Theorem 2.4.3 reads as follows:https://www.physicsforums.com/attachments/5252McInerney leaves the proofs for the Theorem to the reader ...
I am having trouble formulating a proof for Part (3) of the theorem ...
Can someone help...
Homework Statement
Homework Equations
Prof. Note's.
The Attempt at a Solution
I'm on the 3 line where my Prof. combines both equations, I'm confused on what my equation should look. Her's was (n+1)(n+1)+1)/2
Homework Statement
Let R be a ring and suppose r ∈R such that r^2 = 0. Show that (1+r) has a multiplicative inverse in R.
Homework Equations
A multiplicative inverse if (1+r)*x = 1 where x is some element in R.
The Attempt at a Solution
We know we have to use two facts.
1. Multiplicative...
Homework Statement
Suppose R is a commutative ring with only a finite number of elements and no zero divisors. Show that R is a field.
Homework Equations
Unit is an element in R which has a multiplicative inverse. If s∈R with r*s = 1.
A zero divisor is an element r∈R such that there exists...
I am trying to understand the following basic proposition about invertibility: a linear map is invertible if and only if it is injective and surjective.
Now suppose ##T## is a linear map ##T:V\rightarrow W##. The book I read goes the following way in proving the proposition in the direction when...
Prof. S Lakshmi Bala from Department of Physics, Madras, India writes a blackboard of equations which show how beamsplitters used alone affect the wavefunctions of input photons. It seems that it depends on the number of photons you use and in which input port to get you a different entangled...
Hello,
I'm trying to self teach myself Fundamental Mathematics. I looked around, but I wasn't sure what to look for exactly. I read the part on Set Theory in "Book of Proof" by Richard Hammack. I enjoyed it, but I wasn't sure if it is rigorous enough to stand against a college level course...
Homework Statement
Show that every partition of X naturally determines an equivalence relation whose equivalence classes match the subsets from the partition.
Homework Equations
( 1 ) we know that equivalence sets on X can either be disjoint or equal
The Attempt at a Solution
Let Ai be a...
Let G be a subgroup of Sym(X) and ρ ∈ G. Prove that Gρ(x) = ρGxρ−1, where ρGxρ−1 = {ρgρ−1|g ∈ Gx}
What I Know: I need to somehow prove the left is contained in the right and the right is contained in the left.
What I Have Done: Well based on the definition of a stabilizer Gx I assumed that...
I don't know how to Google appropriately for this, since the kind of keywords I use present me with search results that try to define the exponential function using limits instead of what I am trying to ask:
What does the proof look like for the following (assuming f(x) is "nice"). Any sites...
m and n are integers.
log2(i) = m/n
2^(m/n) = i
2^m = i^n
2^0 = i^4 = 1
so that means that log2(i) is rational because there are integers n and m so that log2(i) = m/n , they are m=0 and n=4.
But what I do get about this proof is that it seems to imply that log2(i) = 0/4 = 0 while google says...
Hello Physicsforum
Homework Statement
I have a problem proving this:
Given C(x)=[0, 3/x] for all x\in\chi, with \chi=\Omega being the sample space and P_q=Geom(q) being the geometric distribution.
I have to show that C(x) is a confidence Interval for q but I don't know how to get started...
I'm reading the book by Zee, I came across a paragraph saying that the world is not flat.
"Given an airline table of distances, you can deduce that the world is curved without ever going outside. If I tell you the three distances between Paris, Berlin, and Barcelona, you can draw a triangle on...
Which books in QFT give representations about general proof of renormalization?I know that the book of QFT of Peskin&Schroder does not give the full demontration.
I am trying to prove
##||A||_{\infty} = max_i \sum_{j} |a_{ij}|##
which reads as the ##\infty## norm is the max row sum of matrix A.
##i## is the row index and ##j## is the column index.
Here is what I thought of:
##||A||_{\infty} = sup_{x\neq 0} \frac{||Ax||_{\infty}}{||x||_{\infty}}##
The...
The thought experiment used to prove Lorentz transform uses a light signal as an assumption. What if there was something other than the light signal then Lorentz transformation would have totally different term in place of 'c'(speed of light).
Homework Statement
I am trying to craft a hypothesis regarding factors that affect the coefficient of friction. I know that it is determined by the triboforces and asperity interactions at the interface between the materials (among other factors, but right now I'm just going to focus on this)...
Homework Statement
Successor of a set x is defined as S(x)=x \cup {x}
Prove that if S(x)=S(y) then x=y
Our teacher gives us a hint and says use the foundation axiom.
The Attempt at a Solution
if S(x)=S(y)=x \cup {x}=y \cup {y}
I feel like doing a proof by contradiction would work...
Homework Statement
Show that if a, b, n, m are Natural Numbers such that a and b are relatively prime, then a^n and b^n are relatively prime.
Homework Equations
Relatively prime means 1 = am + bn where a and b are relatively prime. gcd(a,b) = 1
We have a couple corollaries that may be...
Homework Statement
Let f is differentiable function on [0,1] and f^{'}(0)=1,f^{'}(1)=0. Prove that \exists c\in(0,1) : f^{'}(c)=f(c).
Homework Equations
-Mean Value Theorem
The Attempt at a Solution
The given statement is not true. Counter-example is f(x)=\frac{2}{\pi}\sin\frac{\pi}{2}x+10...
Homework Statement
Use the epsilon delta definition to show that lim(x,y) -> (0,0) (x*y^3)/(x^2 + 2y^2) = 0
Homework Equations
sqrt(x^2) = |x| <= sqrt(x^2+y^2) ==> |x|/sqrt(x^2+y^2) <= 1 ==> |x|/(x^2+2y^2)?
The Attempt at a Solution
This limit is true IFF for all values of epsilon > 0, there...
Homework Statement
A. Show that n^n−2/n! < T(n) by looking at how the symmetric group Sn acts on labelled trees. Use |Sn| = n!
T(n) is the number of unlabeled trees on n vertices
Homework EquationsThe Attempt at a Solution
I can't find any mathematical relation between labelled trees and...
Homework Statement
Using the equality ##e = \sum_{k=0}^n \frac{1}{k!} + e^\theta \frac{1}{(n+1)!}## with ##0< \theta < 1##, show the inequality ##0 < n!e-a_n<\frac{e}{n+1}## where ##a_n## is a natural number.
Use this to show that ##e## is irrational.
(Hint: set ##e=p/q## and ##n=q##)...
I've uploaded a proof of the Heisenberg uncertainty principle from Konishi's QM. I just don't quite understand one part: what is the significance of the discriminant being less than or equal to 0? Wouldn't this just result in ## \alpha = R \pm iZ ##? Why would this be desired in this proof?
Homework Statement
Question: Let n> 1 be an integer which is not prime. Prove that there exists a prime p such that p|n and p≤ sqrt(n).
Homework Equations
Fundamental theorem of arithmetic: Every integer n >1 can be written uniquely (up to order) as a product of primes.
The Attempt at a...
Homework Statement
Exercise 0.1. Suppose that G is a finite graph all of whose vertices has degree two or greater. Prove that a cycle passes through each vertex. Conclude that G cannot be a tree.
Homework EquationsThe Attempt at a Solution
If every vertex in a graph G has degree two or...
Homework Statement
Let G be a connected graph. We say that G is minimally connected if the removal of any edge of G (without deleting any vertices) results in a disconnected graph. (a) Show that a connected, minimally connected graph has no cycles. (b) Show that a connected graph with no cycles...
How is the below expression for ##a_{n-2k}## motivated?
I verified that the expression for ##a_{n-2k}## satisfies the recurrence relation by using ##j=n-2k## and ##j+2=n-2(k-1)## (and hence a similar expression for ##a_{n-2(k-1)}##), but I don't understand how it is being motivated.
Source...
Spivak proves that limit of function f (x) as x approaches a is always unique.
ie...If lim f (x) =l
x-> a
and lim f (x) =m
x-> a
Then l=m.
This definition means that limit of function can't approach two different values.
He takes definition of both the limits.
He...
Homework Statement
Prove that a complete graph with n vertices contains n(n − 1)/2 edges.
Homework EquationsThe Attempt at a Solution
The solution gives and inductive proof, but I am just wondering if this works as well.
If we have a set of n vertices or points and we try to match all...
Homework Statement
1. Prove or disprove up to isomorphism, there is only one 2-regular graph on 5 vertices.
Homework EquationsThe Attempt at a Solution
I am making this thread again hence I think I will get more help in this section
old thread...
Homework Statement
1. up to isomorphism, there is only one 2-regular graph on 5 vertices.
Homework EquationsThe Attempt at a Solution
I am still working on the problem, but I don't understand what up to isomorphism means. Does it mean without considering isomorphism?. I just need help with...
Hi All.
I am stuck in a problem.
Please check the image attached.
It's part of a foldable mechanism of a quad copter arm.
The red part is fixed to the body, the grey part is fixed to arms. The transparent part is a threaded collar/sleeve.
The yellow part is a stopper. The hinge is a connecting...
Homework Statement
I work out the problem completely and it does not equal out. Having problems with two variable induction proofs (n and k) in this problem. Below is as far as I got, jpeg below
Homework EquationsThe Attempt at a Solution